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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Second angular derivatives and parabolic iteration in the unit disk


Authors: Manuel D. Contreras, Santiago Díaz-Madrigal and Christian Pommerenke
Journal: Trans. Amer. Math. Soc. 362 (2010), 357-388
MSC (2000): Primary 30D05, 32H40; Secondary 32H50
Published electronically: July 24, 2009
MathSciNet review: 2550155
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Abstract: In this paper we deal with second angular derivatives at Denjoy-Wolff points for parabolic functions in the unit disc. Namely, we study and analyze the existence and the dynamical meaning of this second angular derivative. For instance, we provide several characterizations of that existence in terms of the so-called Koenigs function. It is worth pointing out that there are two quite different classes of parabolic iteration: those with positive hyperbolic step and those with zero hyperbolic step. In the first case, the Koenigs function is in the Carathéodory class but, in the second case, it is even unknown if it is normal. Therefore, the ideas and techniques to approach these two cases are really different. In the end, we also present several rigidity results related to the second angular derivatives at Denjoy-Wolff points.


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Additional Information

Manuel D. Contreras
Affiliation: Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, 41092, Sevilla, Spain
Email: contreras@us.es

Santiago Díaz-Madrigal
Affiliation: Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, 41092, Sevilla, Spain
Email: madrigal@us.es

Christian Pommerenke
Affiliation: Institut für Mathematik, Technische Universität, D-10623, Berlin, Germany
Email: pommeren@math.tu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04873-9
PII: S 0002-9947(09)04873-9
Keywords: Second angular derivative, parabolic functions, Denjoy-Wolff point, Koenigs function, rigidity.
Received by editor(s): December 9, 2005
Received by editor(s) in revised form: February 20, 2008
Published electronically: July 24, 2009
Additional Notes: This research has been partially supported by the Ministerio de Ciencia y Tecnología and the European Union (FEDER) project MTM2006-14449-C02-01 and by La Consejería de Educación y Ciencia de la Junta de Andalucía.
Article copyright: © Copyright 2009 American Mathematical Society